Reduction of warpage of multilayered substrate or package

ABSTRACT

A method that minimizes adjustment of a wiring layer in reducing a warpage of a multilayered substrate and enables location of a part of a wiring layer that needs correction in order to reduce the warpage. The difference in average coefficient of thermal expansion, Δα, varies in a substrate. The method focuses in on the difference in Δα with a great length scale (low frequency) having a relatively significant effect on the warpage compared to the difference in Δα with a smaller length scale (high frequency) and corrects only the difference in Δα with a greater length scale. The distribution of the difference in Δα in a plane of substrate is determined. Then digital filtering is performed to extract only the difference in Δα with a low frequency and the difference in Δα between before and after correction, thereby revealing a part that requires correction.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a warpage of a multilayered substrate in implementation of electronic equipment. More specifically, controlling a warpage of a multilayered substrate in which different materials—a resin substrate material and a wiring material—are distributed two-dimensionally in an upper wiring layer and a lower wiring layer.

2. Description of Related Art

In implementation of electronic equipment, there is a problem of a warpage of a multilayered substrate or package. A warpage of a multilayered organic substrate occurs primarily because of asymmetry of the coefficient of thermal expansion (CTE) between upper and lower wiring layers thereof.

In the upper and lower wiring layers of the multilayered organic substrate, different materials—a resin substrate material and a wiring material—are distributed two-dimensionally.

The resin substrate material is typically an insulating resin, and the wiring material is typically a conductive metal. Therefore, these materials have significantly different coefficients of thermal expansion and elastic moduli, or elasticity, E.

When the wiring material is distributed in the resin substrate material, the wiring material cannot be, inherently, of perfect geometric symmetry, and therefore the distribution of the wiring material is asymmetric. In this respect, there is a demand for a method of optimizing wiring.

According to a known approach to reduce a warpage, a dummy pattern is formed so that the upper and lower wiring layers of the substrate have equal wiring densities. However, at present, there is no way to numerically determine which part should be preferentially corrected in wiring density, and technicians have to empirically or intuitively make the determination.

As a known second best solution, there is a concept of determining an average coefficient of thermal expansion, which is an average of the coefficients of thermal expansion of the upper and lower wiring layers. According to this concept, a warpage of a substrate is analyzed by adjusting the wiring pattern so that the density of the wiring layers is uniform over the whole of the substrate. However, the difference in coefficient of thermal expansion can partially or locally increase, or some wiring patterns cannot be modified for convenience of wiring.

At present, even a partial or local modification of the wiring pattern is usually empirically or intuitively made by the designer of the wiring pattern.

Japanese Patent No. 4204530 describes an FEM analysis that involves dividing an area, in which precise calculation of the warpage is needed, into a fine mesh of elements and dividing an area, in which precise calculation of the warpage is not needed into a coarse mesh of elements.

Japanese Patent No. 3329667 describes a method of determining the position of a reinforcing member with respect to a warpage of a printed circuit board.

PCT Application Publication No. WO 2010/021287 describes a warpage prediction program based on the finite element method (FEM).

Japanese Patent No. 4204524 and Japanese Application Publication Nos. 2012-3460, 2001-14865, 2010-118523, 2010-87145, 2009-260123 describe analyses of a warpage of a substrate based on the finite element method, which can be used for reference purposes.

However, no technique has been found that focuses on variations of the difference in average coefficient of thermal expansion Δα in a substrate and is based on, and applies to a correction, a concept of distinguishing between the difference in coefficient of thermal expansion Δα with a greater length scale (a lower frequency) that has a significant effect on the warpage and the difference in coefficient of thermal expansion Δα with a smaller length scale (a higher frequency) that has a less significant effect on the warpage.

SUMMARY OF THE INVENTION

One aspect of the present invention provides a computer implemented method of performing a calculation associated with controlling warpage of a multilayered substrate. The method includes the steps of (1) discretizing a two-dimensional surface of the multilayered substrate into finite planar elements, where the multilayered substrate has an upper wiring layer and a lower wiring layer, where the upper and lower wiring layers are made of at least two different materials having different elastic moduli, and where the materials are distributed two-dimensionally with different occupancies; (2) calculating a difference in coefficient of thermal expansion of the two-dimensional surface based on composite properties determined from two elastic moduli, two occupancies and the difference between the two occupancies obtained for each element; (3) calculating the warpage of the multilayered substrate based on the calculated composite properties of the two-dimensional surface according to a finite element method; (4) calculating a maximum length scale that corresponds to an allowable warpage; (5) extracting an element involved with a pattern having a length scale greater than the maximum length scale by digital filtering of the difference in coefficient of thermal expansion to produce an extracted element; (6) reducing the difference in coefficient of thermal expansion by adjusting a mixing ratio of the two different occupancies for the extracted element involved with the pattern having a length scale greater than the maximum length scale to produce an adjusted mixing ratio; (7) calculating the warpage of the substrate based on the adjusted mixing ratio of the two different occupancies according to the finite element method; and (8) repeating steps (5), (6) and (7) until the calculated warpage falls below the required warpage.

Another aspect of the present invention provides an information processing system for performing a calculation associated with controlling a warpage of a multilayered substrate. The information processing system includes a memory; a processing device communicatively coupled to the memory; and a calculation performance module communicatively coupled to the memory and processing device, where the calculation performance module is configured to perform a method that includes: (1) discretizing a two-dimensional surface of the multilayered substrate into finite planar elements, where the multilayered substrate has an upper wiring layer and a lower wiring layer, where the upper and lower wiring layers are made of at least two different materials having different elastic moduli, and where the materials are distributed two-dimensionally with different occupancies; (2) calculating a difference in coefficient of thermal expansion of the two-dimensional surface based on composite properties determined from two elastic moduli, two occupancies, and difference between the two occupancies obtained for each finite planar element; (3) calculating the warpage of the multilayered substrate based on the calculated composite properties of the two-dimensional surface according to a finite element method; (4) calculating a maximum length scale that corresponds to an allowable warpage; (5) extracting an element involved with a pattern having a length scale greater than a maximum length scale by digital filtering of the difference in coefficient of thermal expansion to produce an extracted element; (6) reducing the difference in coefficient of thermal expansion by adjusting a mixing ratio of the two different occupancies for the extracted element involved with a pattern having a length scale greater than the maximum length scale to produce an adjusted mixing ratio; (7) calculating the warpage of the multilayered substrate based on the adjusted mixing ratio of the two different occupancies according to the finite element method; and (8) repeating steps (5), (6) and (7) until the calculated warpage falls below a required warpage.

Another aspect of the present invention provides a computer implemented method for performing a calculation associated with controlling warpage in a multilayered substrate, where the multilayered substrate has an upper wiring layer and a lower wiring layer, where the upper and lower wiring layer are each made of at least two different materials that have different elastic moduli, where the materials are distributed two-dimensionally with different occupancies, where a two-dimensional surface of the substrate is discretized into finite planar elements. The method includes the steps of: calculating a difference in coefficient of thermal expansion according to composite properties of the two-dimensional surface determined from two elastic moduli, two occupancies, and difference between the two occupancies obtained for each element; calculating the warpage of the multilayered substrate based on the calculated composite properties of the two-dimensional surface according to a finite element method; extracting an element involved with a pattern having a relatively great length scale by digital filtering of the calculated difference in coefficient of thermal expansion to produce an extracted element; and calculating the difference in coefficient of thermal expansion according to the finite element method based on an adjusted mixing ratio of the two different occupancies for the extracted element involved with a pattern having a length scale greater than a maximum length scale.

Another aspect of the present invention provides a computer readable non-transitory article of manufacture tangibly embodying computer readable instructions which, when executed, cause a computer to carry out the steps of the above methods.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a diagram for illustrating a formula for calculating a warpage (δ) of a multilayered substrate formed by a bimetal.

FIG. 2 is a diagram illustrating that the difference in average coefficient of thermal expansion (Δα) varies in the multilayered substrate.

FIG. 3 shows a graph showing a relationship between a characteristic length scale (L) and a warpage (δ).

FIG. 4 is a flowchart showing the whole of a method according to the present invention.

FIG. 5 is a diagram for illustrating two-dimensional discretization of a material.

FIG. 6 is a diagram for illustrating a calculation of characteristics of a composite material in each element.

FIG. 7 is a diagram for illustrating extraction of patterns having a length scale (L) greater than a maximum length scale (Lmax) by digital filtering of the difference in coefficient of thermal expansion (Δα).

FIG. 8 is a diagram for illustrating correction of patterns having a length scale (L) greater than the maximum length scale (Lmax).

FIG. 9 is a diagram for illustrating calculation of the warpage after the correction.

FIG. 10 is a diagram showing an example of the warpage calculated according to a finite element method (FEM) and actual measurements for comparison.

FIG. 11 is a diagram for illustrating a method of calculating composite body parameters using a multilayered wiring substrate model.

FIG. 12 is a diagram showing a result of the calculation of the composite body parameters using an example of the multilayered wiring substrate model.

FIG. 13 is a diagram showing input values for a multilayered wiring substrate model simulation.

FIG. 14 is a diagram showing a summary of a multilayered beam theory.

FIG. 15 is a diagram showing simultaneous equations of the multilayered beam theory.

FIG. 16 is a diagram for illustrating a method of adding a dummy wiring to a part to be corrected in applying the present invention to a multilayered substrate or package.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

An object of the present invention is to provide an efficient correction method that minimizes adjustment of a wiring layer in reducing a warpage of a multilayered substrate or package. A more specific object of the present invention is to enable the locating of a part of a wiring layer for correction as to effectively reduce a warpage while performing the correction.

The difference in average coefficient of thermal expansion (Δα) varies in a substrate. The present invention focuses on the fact that the difference in coefficient of thermal expansion (Δα) with a great length scale, a low frequency, has a significant effect on the warpage, and the difference in coefficient of thermal expansion (Δα) with a small length scale, a high frequency, has a less significant effect on the warpage. Based on this principle, only the difference in coefficient of thermal expansion (Δα) with a great length scale, or a low frequency, is corrected. The correction is performed by determining the distribution of the difference in coefficient of thermal expansion Δα in a plane of an organic substrate and then determining a part to be corrected by performing digital filtering to extract the difference in coefficient of thermal expansion with a low frequency and determining the difference in coefficient of thermal expansion (Δα) between before and after correction.

The result of the digital filtering that filters out the difference in coefficient of thermal expansion with a high frequency shows a part that requires correction. According to the present invention, correction of a wiring layer performed to reduce a warpage of a substrate can be minimized, and correction can be performed on only a part of the substrate that requires correction.

FIG. 1 is a diagram for illustrating a formula for calculating a warpage (δ) of a multilayered substrate formed by a bimetal. The formula is well known as a formula that expresses a warpage of a bimetal in the field of strength of materials. The multilayered substrate has an upper wiring layer and a lower wiring layer. Each wiring layer is composed of at least two different materials—a resin substrate material, m, and a wiring material, f—having different elastic moduli (E), which are distributed two-dimensionally with different occupancies (V). This diagram shows four different materials distributed. As can be seen, the warpage (δ) is proportional to the difference in coefficient of thermal expansion (Δα).

FIG. 2 is a diagram illustrating that the difference in average coefficient of thermal expansion (Δα) varies in the multilayered substrate. As shown by the graph, a warpage occurs because of the difference in coefficient of thermal expansion (Δα) between the upper and lower wiring layers. The difference in coefficient of thermal expansion (Δα) has a pattern (L1) that has a relatively small length scale (L) and a pattern (L2) that has a relatively great length scale (L).

According to the present invention, it is enough to correct only a large undulation that corresponds to the pattern (L2) that has a relatively great length scale (L). In addition, a characteristic length Lmax for an allowance warpage (δmax) can be determined according to the formula for a bimetal shown in FIG. 1.

FIG. 3 shows a graph showing a relationship between the characteristic length (L) and the warpage (δ). If the allowable warpage (δmax) is 5 μm, the characteristic length L is 4.2 mm. Various other values of the allowable warpage (δmax) can be specified by taking into consideration safety factors or other parameters. The allowable warpage (δmax) can be set as a threshold for determining whether a finite element method (FEM) simulation can be ended or not. The allowable warpage (δmax) can be automatically calculated by a computer or input to a computer to be used for a computation.

FIG. 4 is a flowchart showing each step of the method according to the present invention. As shown, during each step, a process is performed. For purposes of clarity, each process is given a number in the flowchart and thereby corresponds to the numbers of the subsequent drawings.

(1) Two-dimensional discretization of the substrate corresponds to FIG. 5.

(2) Calculation of characteristics of a composite material in each element corresponds to FIG. 6.

(3) Calculation of the warpage δ according to the finite element method corresponds to FIG. 9.

(4) Calculation of the maximum length scale (Lmax) corresponding to the allowable warpage (δmax) corresponds to FIG. 2, 3, 12, 14 or 15.

(5) Extraction of a pattern having a length scale greater than the maximum length scale (Lmax) by filtering of the difference in coefficient of thermal expansion Δα corresponds to FIG. 7.

(6) Correction of the pattern having a length scale greater than the maximum length scale (Lmax) corresponds to FIG. 8.

(7) Calculation of the warpage δ according to the finite element method after the correction corresponds to FIG. 9.

(8) It is determined whether the calculated value of the warpage after the correction satisfies the required condition: δ<δmax.

(9) The maximum length scale (Lmax) is adjusted.

(10) Ending of optimization corresponds to FIG. 10.

That is, the method according to the present invention includes the steps described below.

(1) A step of discretizing the two-dimensional surface of the substrate into finite planar elements.

(2) A step of calculating the difference in coefficient of thermal expansion (Δα) of the two-dimensional surface based on the composite properties determined from two elastic moduli (E), two occupancies (V), and the difference between the two occupancies (V) obtained for each element.

(3) A step of calculating the warpage (δ) of the substrate based on the calculated composite properties of the two-dimensional surface according to the finite element method (FEM).

(4) A step of calculating the maximum length scale (Lmax) that corresponds to the allowable warpage (δmax).

(5) A step of extracting an element involved with a pattern having a length scale greater than the maximum length scale (Lmax) by digital filtering of the difference in coefficient of thermal expansion (Δα).

(6) A step of reducing the difference in coefficient of thermal expansion (Δα) by adjusting the mixing ratio of the two different occupancies (V) for the element involved with the pattern (2L) having a length scale (L) greater than the maximum length scale (Lmax).

(7) A step of calculating the warpage (δ) of the substrate based on the adjusted mixing ratio of the two different occupancies (V) according to the finite element method (FEM).

(8) A step of repeating the steps (5), (6) and (7) until the calculated warpage (δ) falls below the allowable warpage (δmax).

FIG. 5 is a diagram for illustrating two-dimensional discretization of a material. The finite element method is used herein. The finite element method (FEM) is a numerical analysis approach for numerically determining an approximate solution of a differential equation that is difficult to analytically solve. The finite element method involves dividing a domain in which an equation is defined into smaller regions, or elements, and determining a relatively simple common interpolation function that approximates to the equation in each smaller region.

The finite element method was first developed in the field of structural mechanics and has become widely used in other fields. The theory behind the finite element method is associated with functional analysis and mathematically systematic.

In the case of a static analysis, provided that a complicated structure is regarded as an aggregate of small elements, a linear equation is solved by substituting a boundary condition, i.e., Dirichlet condition or Neumann boundary condition, of displacement at each node into the linear equation.

Structural analysis of a deformation of a target structure caused by an external force, for example, is generally achieved in a displacement method, in which displacement is an unknown quantity, or a stress method, in which stress is an unknown quantity. The structural analysis based on the finite element method is typically achieved in the displacement method.

According to the present invention, which is intended to calculate the warpage, the displacement method is used. In step (1), the two-dimensional surface of the substrate to be simulated is discretized into finite planar elements. In the example shown in this drawing, the two-dimensional surface is discretized into 100 (10×10) finite elements in total, 10 elements 1 to 10 in an i direction by 10 elements 1 to 10 in a j direction.

A parameter that changes in a finite element is typically prepared as an array for containing a variable in a software program, and the simulation is carried out through cooperation of many memory areas, or hardware resources.

In the description herein, the simulation method is described as being based on the finite element method (FEM). However, a wide variety of other numerical analysis methods, such as the boundary element method and the difference method, can be applied. In this respect, the phrase or expression “calculate according to the finite element method (FEM)” should be construed broadly.

FIG. 6 is a diagram for illustrating a calculation of characteristics of the composite material in each element. In step (2), characteristics of the composite material in each element, that is, composite properties of the two-dimensional surface can be calculated according to a formula that is well-known in the field of strength of materials.

FIG. 7 is a diagram for illustrating extraction of patterns having a length scale (L) greater than the maximum length scale (Lmax) by digital filtering of the difference in coefficient of thermal expansion (Δα). By digital filtering of the difference in coefficient of thermal expansion (Δα), elements involved with the patterns having a relatively great length scale can be extracted. These patterns can be visualized as shown in the drawing.

The digital filtering of the difference in coefficient of thermal expansion (Δα) performed for extracting elements involved with patterns having a relatively great length scale in step (5) can be a digital filtering performed after a fast Fourier transform, and an inverse fast Fourier transform can be performed after the digital filtering. Although the processing based on the fast Fourier transform has an affinity for the digitized discretized processing, the applicability of the present invention is not limited thereto.

FIG. 8 is a diagram for illustrating correction of patterns having a length scale (L) greater than the maximum length scale (Lmax). The correction means adjustment of the mixing ratio of the two different occupancies (V) for the elements involved with the patterns (L2) whose extracted length scale (L) is relatively great.

The correction can be carried out in at least three optional ways. First, the correction can be an adjustment that involves adding a correcting dummy wiring layer to the part of the pattern. Second, the correction can be an adjustment of the wiring density of the wiring material f in a part of the wiring layer that is not significantly affected by the adjustment. Third, the correction can be an adjustment of the wiring pattern itself in the wiring layer.

After the correction, the calculation according to the finite element method (FEM) can be performed again. The part to be corrected can be automatically recognized and calculated by a computer or can be input to a computer for computation through manual adjustment by a user.

FIG. 9 is a diagram for illustrating calculation of the warpage after the correction. FIG. 10 is a diagram showing an example of the warpage calculated according to the finite element method (FEM) and actual measurements for comparison.

As can be seen from the diagram, the calculated warpage tends to agree with the actual measurements.

FIGS. 11 to 13 illustrate a multilayered wiring substrate model. An average volume ratio of a Cu (copper) wiring can be used, and such a model can be used for determining the characteristic length scale for the allowable warpage.

FIGS. 14 and 15 depict a known model that can also be used for determining the characteristic length scale for the allowable warpage. As described above, the characteristics of the mixed material can be calculated in various methods.

FIG. 16 is a diagram for illustrating a method of adding a dummy wiring to a part to be corrected in applying the present invention to a multilayered substrate or package. This is an application as an implementation package for electronic equipment. The wiring can be corrected by placing a dummy wiring at a part having a great length scale. 

We claim:
 1. A computer implemented method of performing a calculation associated with controlling warpage in a multilayered substrate, the method comprising the steps of: (1) discretizing a two-dimensional surface of the multilayered substrate into finite planar elements, wherein the multilayered substrate has an upper wiring layer and a lower wiring layer, wherein the upper and lower wiring layers are made of at least two different materials having different elastic moduli, and wherein the materials are distributed two-dimensionally with different occupancies; (2) calculating a difference in coefficient of thermal expansion of the two-dimensional surface based on composite properties determined from two elastic moduli, two occupancies, and difference between the two occupancies obtained for each finite planar element; (3) calculating the warpage of the multilayered substrate based on the calculated composite properties of the two-dimensional surface according to a finite element method; (4) calculating a maximum length scale that corresponds to an allowable warpage; (5) extracting an element involved with a pattern having a length scale greater than a maximum length scale by digital filtering of the difference in coefficient of thermal expansion to produce an extracted element; (6) reducing the difference in coefficient of thermal expansion by adjusting a mixing ratio of the two different occupancies for the extracted element involved with a pattern having a length scale greater than the maximum length scale to produce an adjusted mixing ratio; (7) calculating the warpage of the multilayered substrate based on the adjusted mixing ratio of the two different occupancies according to the finite element method; and (8) repeating steps (5), (6) and (7) until the calculated warpage falls below a required warpage.
 2. The method according to claim 1, wherein the digital filtering of step (5) is performed after a fast Fourier transform.
 3. The method according to claim 1, wherein an inverse fast Fourier transform is performed after the digital filtering.
 4. The method according to claim 1, wherein step (6) of adjusting a mixing ratio of the two different occupancies for the extracted element involved with the pattern having a length scale greater than the maximum length scale is an adjustment that involves adding a dummy wiring layer.
 5. The method according to claim 1, wherein step (6) of adjusting a mixing ratio of the two different occupancies for the extracted element involved with the pattern having a length scale greater than the maximum length scale is an adjustment of a wiring density of a wiring material in a wiring layer.
 6. The method according to claim 1, wherein step (6) of adjusting a mixing ratio of the two different occupancies for the extracted element involved with a pattern having a length scale greater than the maximum length scale is an adjustment of a wiring pattern in a wiring layer.
 7. An information processing system for performing a calculation associated with controlling a warpage of a multilayered substrate, the information processing system comprising: a memory; a processing device communicatively coupled to the memory; and a calculation performance module communicatively coupled to the memory and processing device, wherein the calculation performance module is configured to perform a method comprising the steps of: (1) discretizing a two-dimensional surface of the multilayered substrate into finite planar elements, wherein the multilayered substrate has an upper wiring layer and a lower wiring layer, wherein the upper and lower wiring layers are made of at least two different materials having different elastic moduli, and wherein the materials are distributed two-dimensionally with different occupancies; (2) calculating a difference in coefficient of thermal expansion of the two-dimensional surface based on composite properties determined from two elastic moduli, two occupancies, and difference between the two occupancies obtained for each finite planar element; (3) calculating the warpage of the multilayered substrate based on the calculated composite properties of the two-dimensional surface according to a finite element method; (4) calculating a maximum length scale that corresponds to an allowable warpage; (5) extracting an element involved with a pattern having a length scale greater than a maximum length scale by digital filtering of the difference in coefficient of thermal expansion to produce an extracted element; (6) reducing the difference in coefficient of thermal expansion by adjusting a mixing ratio of the two different occupancies for the extracted element involved with a pattern having a length scale greater than the maximum length scale to produce an adjusted mixing ratio; (7) calculating the warpage of the multilayered substrate based on the adjusted mixing ratio of the two different occupancies according to the finite element method; and (8) repeating steps (5), (6) and (7) until the calculated warpage falls below a required warpage.
 8. The system according to claim 7, wherein step (5) of extracting an element involved with a pattern having a length scale greater than the maximum length scale by digital filtering of the difference in coefficient of thermal expansion is the digital filtering performed after a fast Fourier transform, and then an inverse fast Fourier transform is performed.
 9. A computer readable non-transitory article of manufacture tangibly embodying computer readable instructions which, when executed, cause a computer to carry out the steps of a method according to claim
 1. 10. A computer implemented method for performing a calculation associated with controlling warpage in a multilayered substrate, wherein the multilayered substrate has an upper wiring layer and a lower wiring layer, wherein the upper and lower wiring layer are each made of at least two different materials that have different elastic moduli, wherein the materials are distributed two-dimensionally with different occupancies, wherein a two-dimensional surface of the substrate is discretized into finite planar elements, the method comprising the steps of: calculating a difference in coefficient of thermal expansion according to composite properties of the two-dimensional surface determined from two elastic moduli, two occupancies, and difference between the two occupancies obtained for each element; calculating the warpage of the multilayered substrate based on the calculated composite properties of the two-dimensional surface according to a finite element method; extracting an element involved with a pattern having a relatively great length scale by digital filtering of the calculated difference in coefficient of thermal expansion to produce an extracted element; and calculating the difference in coefficient of thermal expansion according to the finite element method based on an adjusted mixing ratio of the two different occupancies for the extracted element involved with a pattern having a length scale greater than a maximum length scale.
 11. The computer implemented method of claim 10, wherein the method further comprises the step of calculating a maximum length scale that corresponds to an allowable warpage.
 12. A computer readable non-transitory article of manufacture tangibly embodying computer readable instructions which, when executed, cause a computer to carry out the steps of a method according to claim
 10. 